Properties

Label 1-6145-6145.112-r1-0-0
Degree 11
Conductor 61456145
Sign 0.139+0.990i-0.139 + 0.990i
Analytic cond. 660.371660.371
Root an. cond. 660.371660.371
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯
L(s)  = 1  + (−0.617 + 0.786i)2-s + (−0.859 − 0.511i)3-s + (−0.238 − 0.971i)4-s + (0.932 − 0.360i)6-s + (−0.848 − 0.529i)7-s + (0.911 + 0.412i)8-s + (0.476 + 0.879i)9-s + (0.895 + 0.444i)11-s + (−0.292 + 0.956i)12-s + (0.826 − 0.563i)13-s + (0.940 − 0.340i)14-s + (−0.886 + 0.462i)16-s + (−0.984 − 0.178i)17-s + (−0.985 − 0.168i)18-s + (0.992 + 0.122i)19-s + ⋯

Functional equation

Λ(s)=(6145s/2ΓR(s+1)L(s)=((0.139+0.990i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(6145s/2ΓR(s+1)L(s)=((0.139+0.990i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 61456145    =    512295 \cdot 1229
Sign: 0.139+0.990i-0.139 + 0.990i
Analytic conductor: 660.371660.371
Root analytic conductor: 660.371660.371
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ6145(112,)\chi_{6145} (112, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 6145, (1: ), 0.139+0.990i)(1,\ 6145,\ (1:\ ),\ -0.139 + 0.990i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.3188412047+0.3667441261i0.3188412047 + 0.3667441261i
L(12)L(\frac12) \approx 0.3188412047+0.3667441261i0.3188412047 + 0.3667441261i
L(1)L(1) \approx 0.5316084831+0.06003288065i0.5316084831 + 0.06003288065i
L(1)L(1) \approx 0.5316084831+0.06003288065i0.5316084831 + 0.06003288065i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
1229 1 1
good2 1+(0.617+0.786i)T 1 + (-0.617 + 0.786i)T
3 1+(0.8590.511i)T 1 + (-0.859 - 0.511i)T
7 1+(0.8480.529i)T 1 + (-0.848 - 0.529i)T
11 1+(0.895+0.444i)T 1 + (0.895 + 0.444i)T
13 1+(0.8260.563i)T 1 + (0.826 - 0.563i)T
17 1+(0.9840.178i)T 1 + (-0.984 - 0.178i)T
19 1+(0.992+0.122i)T 1 + (0.992 + 0.122i)T
23 1+(0.262+0.964i)T 1 + (-0.262 + 0.964i)T
29 1+(0.267+0.963i)T 1 + (-0.267 + 0.963i)T
31 1+(0.997+0.0715i)T 1 + (-0.997 + 0.0715i)T
37 1+(0.5420.840i)T 1 + (-0.542 - 0.840i)T
41 1+(0.2570.966i)T 1 + (0.257 - 0.966i)T
43 1+(0.1020.994i)T 1 + (-0.102 - 0.994i)T
47 1+(0.471+0.881i)T 1 + (-0.471 + 0.881i)T
53 1+(0.8740.485i)T 1 + (-0.874 - 0.485i)T
59 1+(0.3160.948i)T 1 + (-0.316 - 0.948i)T
61 1+(0.355+0.934i)T 1 + (0.355 + 0.934i)T
67 1+(0.995+0.0970i)T 1 + (0.995 + 0.0970i)T
71 1+(0.8110.584i)T 1 + (0.811 - 0.584i)T
73 1+(0.9930.117i)T 1 + (-0.993 - 0.117i)T
79 1+(0.4120.911i)T 1 + (-0.412 - 0.911i)T
83 1+(0.902+0.430i)T 1 + (0.902 + 0.430i)T
89 1+(0.502+0.864i)T 1 + (-0.502 + 0.864i)T
97 1+(0.7080.705i)T 1 + (-0.708 - 0.705i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.28960404945270653494656184204, −16.7432328261993722558879486101, −16.149075501203762003627431177402, −15.783164444862716838209786518561, −14.852781268536110971843169516131, −13.844916720169518154213061945149, −13.17152455026543043647120709739, −12.53340290858164192301667512242, −11.819035145908835340169454094573, −11.33148846443576596578171476127, −10.92109875061256624496643620613, −9.87289826414094611350708807046, −9.57764019030100845315928475854, −8.87728836202056384493894480102, −8.3471727030478056249234367807, −7.115431033564388231073634824470, −6.45885195091855685891647098748, −6.03156799903094476733160584901, −4.928296047526315639605139509906, −4.14048505912964493531973578515, −3.60759388039864053170759170835, −2.87828828158501801664914880916, −1.82571352057156616739922343920, −1.02356240101523553528259066847, −0.17292227733055699202221612669, 0.55118987845516665875860962346, 1.350104307308432741280074583303, 1.970070981788753322495566837004, 3.43137829047396332527436890038, 4.11449132750444796119879136132, 5.13882415536515094913888696924, 5.68080332935908459292041221218, 6.367628427480199399700891873360, 7.05153621460056972161937124060, 7.295913197994477803106705690216, 8.19712820420642361280045909153, 9.21102022893066886528764237766, 9.505790926391438988503809184782, 10.514960463867944534659297262962, 10.89492215610008961896958294954, 11.63471087428712945524509205467, 12.545755987483435851061802823076, 13.15263919940054327389035459419, 13.80515232582336969800844408197, 14.34671960495216765280887850065, 15.456808118348257969057433355670, 15.95422660356030296409239090124, 16.33502108248296329685666899359, 17.14336372532997795327607754175, 17.69980564620499015775483743123

Graph of the ZZ-function along the critical line